text stringlengths 1 2.12k | source dict |
|---|---|
By this I am getting relation as --
R= A / (root 3 * 2) which is not the same
Once you have got R= A / (root 3 * 2)
You need to rationalize the denominator, or simply remove the under root from the denominator
Multiply $$\sqrt{3}$$ on both numerator and denominator and you will get
R= A $$\sqrt{3}$$/ 6
Does this hel... | {
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Thanks
Kudos [?]: [0], given: 41
Re: If a circle is inscribed in an equilateral triangle, what is the area [#permalink] 23 Aug 2017, 08:41
Similar topics Replies Last post
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Topics:
1 A circle is inscribed in equilateral triangle XYZ, which is itself 2 14 Sep 2017, 22:01
41 Circle O is inscribed in equilatera... | {
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1. ## Re: sketching curve graph
Okay good, you have found:
x-intercepts: (-2,0), (-1,0), (1,0)
y-intercept: (0,-2)
Now, can you answer parts a) - c) above? Can you use both the first and second derivative tests to clearly show what the nature of the extrema are? While one one of these test is sufficient, I think is... | {
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https://www.dropbox.com/s/v3v51fukz5...2012.52.01.jpg
7. ## Re: sketching curve graph
here's what i did for the final graph...includes all my work. not sure about concavity though
https://www.dropbox.com/s/7p5hq991lc...2013.39.34.jpg
https://www.dropbox.com/s/iibkdwpc0j...2013.39.44.jpg
https://www.dropbox.com/s/u91s... | {
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$f''(x)=6x+4=0$
Solving for $x$, we find the critical value is at:
$x=-\frac{2}{3}$
Since the second derivative is an increasing linear function, we know then that:
On the interval:
$\left(-\infty,-\frac{2}{3} \right)$ we find $f(x)$ is concave down.
$\left(-\frac{2}{3},\infty \right)$ we find $f(x)$ is concave u... | {
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$f(x)=x^3+2x^2-x-2=x^2(x+2)-(x+2)=$
$(x^2-1)(x+2)=(x+2)(x+1)(x-1)=0$
So, we know the $x$-intercepts are at:
$(-2,0),\,(-1,0),\,(1,0)$
To find the $y$-intercept, we let $x=0$ and find:
$f(0)=-2$
and so we know the $y$-intercept is at:
$(0,-2)$
Putting all of this together, we should obtain a graph that looks lik... | {
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Two dice - probability that sum of scores > 9 or individual scores differ by 1.
• November 20th 2011, 06:04 PM
Punch
Two dice - probability that sum of scores > 9 or individual scores differ by 1.
Two fair dice are thrown together, and the scores added. What is the probability that the total score is 9, or the individ... | {
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There are $12$ desired outcomes.
The probability is:. $\frac{12}{36} \,=\,\frac{1}{3}$
• November 20th 2011, 10:35 PM
takatok
Re: Two dice - probability that sum of scores > 9 or individual scores differ by 1.
While enumerating every possibility and finding the answer by counting is possible, it can become unwieldy f... | {
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# Optimizing computational algorithms
Karol Dowbecki · May 29, 2021
When optimizing an algorithm it’s often good to take a step back and think about the problem domain before jumping straight to coding. Today I have seen following question asked on Stack Overflow:
I have a program that has two nested for loops and t... | {
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If we apply these summation formulas to our equation:
$b \sum\limits_{k = 0}^{b-1} k - \sum\limits_{k = 0}^{b-1} k^2 = b \frac{(b-1) (b-1+1)}{2} - \frac{(b-1) (b-1+1) [2(b-1) + 1]}{6} = \newline = b \frac{(b-1) b}{2} - \frac{(b-1) b (2b-1)}{6} = \frac{b^3- b^2}{2} - \frac{(b^2-b) (2b-1)}{6} = \newline = \frac{b^3- b^2... | {
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# Will there at some point be more numbers with $n$ factors than prime numbers for any $n$? [duplicate]
Let $$\pi(x)$$ be the prime counting function: the number of numbers $$\leq x$$ with just one prime factor. Let $$\pi_n(x)$$ count the number of numbers $$\leq x$$ with exactly $$n$$ prime factors (counted with mult... | {
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Interesting question. My intuition is that $$\pi_n(x)>\pi(x)$$ for large $$x$$.
Heuristic argument: We note that, for each $$n$$, the sum of the reciprocals of the $$n-$$primes diverges. (here, of course, an $$n-$$prime means a natural number with exactly $$n$$ prime divisors). This is clear since we can just pick som... | {
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• just naively, the number of semiprimes up to $n$ is always greater than $({\sqrt{n}\over \ln\sqrt{n}})^2$ – user645636 Feb 4 at 1:00
• @RoddyMacPhee Again, you mean for large enough $n$. And you can get a similar formula for $n-$primes. But I think that you get sharper bounds out of terms like $\lambda x$ then out of... | {
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Problem
Prove or disprove that, for any $n \in \mathbb{N_+}$, there exist $a,b \in \mathbb{N_+}$ such that $$\frac{a^2+b}{a+b^2}=n.$$
My Thought
Assume that the statement is ture. Then, the equality is equivalent to that
$$a^2-na+b-nb^2=0.$$
Regard it as a quadratic equation with respect of $a$.Then $$a=\dfrac{n \... | {
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$$n(2a-n)^2-(2nb-1)^2=n^3-1,$$
which is a $\textbf{ Pell-like equation}$. This will help?
• Computer search confirms the conjecture for $1 \leq n \leq 100$, though the requisite values of $a$ and $b$ can be very large (for example, $n=54$ gives $(a,b) = (9\,683\,509, 1\,317\,755)$ and $n=89$ gives $(a, b) = (22\,276\... | {
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If you replace $x_1^2=ny_1^2+1$ into the expression for $x_2$ you get:
$$x_2=1+2ny_1^2\implies x_2\equiv1\space (\text{mod}\space 2n)$$
This also proves that $x_2$ has to be odd (which makes perfect sense because solutions of Pell's equation are always co-prime and $y_2$ is even).
You can construct more solutions of... | {
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For example:
ABPair[5613]
{60584278414870816497213, 808653403020126409200, 5613}
The third number is just a check that the calculated numbers are valid. In other words:
$$\frac{60584278414870816497213^2+808653403020126409200}{60584278414870816497213+808653403020126409200^2}=5613$$
The script is lightning fast even... | {
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Choose \begin{align} a=\frac{k^2(k^3+2)}{4}, b=\frac{k^4}{4}. \end{align}
Conditions imply that $$k^2 \equiv 0 \pmod {4}$$ and so both $$a$$ and $$b$$ are integers. By algebraic manipulation we can show that $$(a^2+b)/(b^2+a)=k^2=n$$ (it is quite technical).
Case 2: $$n=k^2,k \equiv 1 \pmod {2}$$
Choose
\begin{alig... | {
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# Counting overlapping figures
How many four-sided figures appear in the diagram below?
I tired counting all the rectangles I could see, but that didn't work. How do I approach this?
• Just count it!:D – Mahdi Jul 20 '14 at 11:46
• In what sense did "counting all the rectangles" not work? Do you feel that you miscou... | {
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If the leftmost edge is 2: Then the rightmost edge is 3 or 4 (2 choices), and in either case there are 3 horizontal segments that can serve as the top/bottom($\binom{3}{2} =3$ choices). So this gives $2 \cdot 3 = 6$. 6 options.
If the leftmost edge is 3: If the rightmost edge is 5 there is only one rectangle. If the r... | {
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# Math-Probability
Suppose an urn contains 8 red,5 white and 7 blue marbles.If 3 marbles are drawn at random from the urn with replacement,what is the probability that three marbles are the same color?Give answer in a reduced fraction.
1. could be RRR , WWW, or BBB
prob = (8/20)(7/19)(6/18) + (5/20)(4/19)(3/(18) + (... | {
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Three marbles are drawn without replacement from an urn containing 4 red marbles, five what marbles, and two blue marbles. Determine the probability that None is red
8. ### Finite Math
An urn contains 8 blue marbles and 7 red marbles. A sample of 6 marbles is chosen from the urn without replacement. What is the probab... | {
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# Math Help - Combinatio Help
1. ## Combinatio Help
Suppose that a club consists of 10 men and 13 women. The club is going to form a committee of 8 people. How many 8 person committees have more women than men?
I thought maybe it was (23!/8!*15!)-(10!/8!*2!) but that does not seem like it gives me the correct answer... | {
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Check . There are: . ${23\choose8} \;=\;{\color{blue}490,\!314}$ possible committees . . . YAY!
3. I see what you did there. I was wayyyyyyy off! But I now understand how you came about that answer. The formula using summation notation would be (I wish I could say I came up with that solely on my own).
$\sum\limits_{... | {
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# Curves with a common tangent line
• Question
Find the point where the curves $$\tag 1y = x^3 -3x + 4$$ and $$\tag 2 y = 3x^2 - 3x$$ are tangent to each other, that is, have a common tangent line.
• My approach
• Let $x = a$ and $x = b$ be the points on curves $(1)$ and $(2)$, respectively, at which their slopes a... | {
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Any suggestions?
• Does "have a common tangent line" require that the two curves intersect each other as well? We can imagine a line that is tangent to both curves without the points of tangency being identical. – Rory Daulton Nov 10 '14 at 22:38
• @RoryDaulton that's what caused me some confusion, glad you ask. Prior... | {
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Two curves $y = f(x)$ and $y = g(x)$ have a common tangent line at $x = a$ iff:
• They intersect there: $f(a) = g(a)$.
• Their tangent lines have equal slope there: $f'(a) = g'(a)$.
Since quadratic equations are easier to solve than cubic ones, we start with the second condition: $$3a^2 - 3 = 6a - 3 \iff a^2 - 2a = 0 \... | {
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# Math Help - calcii: contours
1. ## calcii: contours
Hi, I am having trouble with contours, is there a procedure to do these?
the question I am working on is when z=c
z=((x^2)+(y^2))/2x
what i tried to do:
c2x=(x^2)+(y^2)
c2x-x^2=y^2
x(2c-x)=y^2
2c-x=(y^2)/x
2c=((y^2)/x)+x
and that is where I got stuck. I cant ... | {
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You can also look at this as a pair of right elliptical cones, each with a vertex at (0, 0, 0) and axis along the line $z={4\over3}x$ in the x-z plane. (I haven't figured out the eccentricity of the ellipses.)
5. $c= \frac{x^2+ y^2}{2x}$ gives $2cx= x^2+ y^2$, $x^2- 2cx+ y^2= 0$
Now "complete the square": $x^2- 2cx+ c... | {
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Parity
Definition
Parity is a term we use to express if a given integer is even or odd. The parity of a number depends only on its remainder after dividing by $2$. An even number has parity $0$ because the remainder after dividing by $2$ is $0$, while an odd number has parity $1$ because the remainder after dividing ... | {
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When posting on Brilliant:
• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" ... | {
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# Relationship Between Volume Of Cylinder Cone And Sphere | {
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So let's see what that would look like if we apply it to the surface areas. Continue to reduce the size of the spheres, and you approach the 74% figure of "ideal packing". This chart type also includes cylinder, cone, and pyramid subtypes. Cylinder examples/objects Colored paper Calculator Beans Scissors Copies of T870... | {
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the area of the sphere in ndimensions=A(n-1). As nouns the difference between cone and cylinder is that cone is (label) a surface of revolution formed by rotating a segment of a line around another line that intersects the first line while cylinder is (geometry) a surface created by projecting a closed two-dimensional ... | {
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is 8 / c m. STAAR ALGEBRA I REFERENCE MATERIALS. The volume of a hyperspherical cone V n cone is also easy to derive by the difference between the sector volume and the cap volume, V n cone (r) = V n sector (r)-V n cap (r) = 1/nV n-1 (rsinφ)rcosφ. Which of the following is true? A The volumes are the same. Comparing Cy... | {
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cm 3. Example 6: Find the formula for the total surface area of each figure given bellow :. It takes three cones full of rice to fill the cylinder. 86mm) and, after skull trepanation, a post-surgical CT (512 sagittal, 635 coronal and 68 axial slices, voxel-size of 0. vol of a cylinder = 3 * vol of a cone. Solve for the ... | {
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two decimal places. Solve real‐world and mathematical problems involving volume of cylinders, cones, and spheres. I know that the volume we're interested in is the volume of the intersection between the sphere of Stack Exchange Network Stack Exchange network consists of 177 Q&A communities including Stack Overflow , th... | {
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is three times the volume of the cone. In this section, we consider several problems in which two or more related quantities are changing and we study how to determine the relationship between the rates of change of these quantities. What is the ratio of the cone’s height to its radius? (2003 AMC 12B Problems. Since a ... | {
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between the change in dimensions and the resulting change in surface area and volume. 0 Equation Volume of a Cylinder, Cone, and Sphere Volume Cylinder Previous Formulas Learned Area and Circumference of a Circle Cylinder Volume of a Cylinder Volume of a Cylinder Volume of Cylinders Class Practice Volume of Cylinders. ... | {
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is that cone is (label) a surface of revolution formed by rotating a segment of a line around another line that intersects the first line while cylinder is (geometry) a surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve. 72 in3 C) 392. Surface area to volume r... | {
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is equal to because the base is shaped like a circle. The bottom of the cylinder will be on the z = 0 {\displaystyle z=0} plane for simplicity of calculations. Cones and Cylinders. This volume formula applies to all cones, including oblique cones. Therefore, at every height the slice of area in the cylinder intersectio... | {
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the cylinder? Make a conjecture and try to convince other students. As a first example we study the influence of head tissue conductivity inhomogeneity. Sphere: A solid figure that has all points the same distance from the center. STAAR ALGEBRA I REFERENCE MATERIALS. Calculate volume of a cone if you know radius and he... | {
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dV dt = 400 πr. For example, a sphere represents a shape that has the highest volume to surface area ratio. Odd-shaped objects You can find the volume of an odd-shaped object, like a key, by placing it in water. The area of a circle. com/watch?v=3wuJJqlr6m0 To find manipulatives similar to these, try looking. vol of a ... | {
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the relationship between the diameter of a sphere and its volume. The radius of a sphere is 5 yards. Their volumes can easily be seen to be (4/3) r 3, 2(1/3) r 3, and 2 r 3. For the polar or normal aspect, the cone. Lesson Notes Students informally derive the volume formula of a sphere in Lesson 12 (G-GMD. 70 in3 D) 52... | {
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MULTIPLE CHOICE Let V be the volume of a sphere, S be the surface area of the sphere, and r be the radius of the sphere. Volume Find the volume of a sphere that has Ch. Note that we can assume $$z$$ is positive here since we know that we have the upper half of the cone and/or sphere. Convert between weight and volume u... | {
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and Volume Purpose: For this activity you will be performing a few measurements to help describe the relationship between mass and volume. For example, after part (b), the teacher could ask the students for other ways to determine which vase holds the most water, with the expectation that students might respond with. E... | {
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a sphere is an object. 08 cubic centimeters. Processing. Click here to check your answer to Practice Problem 3. Fill the cone with rice, then pour the rice into the cylinder. If B is the area of the base of a pyramid or a cone and H is the height of the solid, then the formula for the volume if V=1/3 BH. Big Ideas: Vol... | {
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two points along the sphere, we produce a radial projection of the polyhedron. Work out the slant height of the cone to 1dp. All that is left is to calculate the area of the sphere in ndimensions=A(n-1). Surface Area is the area of the outer part of any 3D figure and Volume is the capacity of the figure i. Solved Probl... | {
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and millimeters. and surface to volume ratio of a frustum of right circular cone Definition of a frustum of a right circular cone : A frustum of a right circular cone (a truncated cone) is a geometrical figure that is created from a right circular cone by cutting off the tip of the cone perpendicular to its height H. N... | {
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of the hemisphere first. Assessment Handbook, p. Areaand&Volume& JimKing University&of&Washington& NWMI2013& Cavalieri&for&Area • 2Hdimensional&case:&Suppose&two®ions&in&aplane&are& included&between&two¶llel&lines&in&thatplane. The volume tells us something about the capacity of a figure. 05 in3 B) 104. The volume of a... | {
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same shape as the earth, it shows sizes and shapes more accurately than a mercator projection map (a flat representation of the earth). Videos / Movies •Friction loss and analysis. All these surfaces are related and can easily slip from one to another. Design #1 is a hemisphere hollowed out of a cylinder, and design #2... | {
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cylinder when they both have the same radius and height? Volume of Cylinder and Cone. Cones Volume = 1/3 area of the base x height V= r2h Surface S = r2 + rs. 8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. open top height of height of empty space cylinder height. Downloa... | {
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one circular base face and one continuous curved top face. To do so, they examine the relationship between a hemisphere, cone, and cylinder, each with the same radius, and for the cone and cylinder, a height equal to the radius. Use Pythagoras' theorem to find a relationship between r 2 and h 2. 847 KEY VOCABULARY Now ... | {
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and a sphere equal to the inner. Relationship between volume of a pyramid and prism - Duration: Easiest way to Learn Volume of Cylinder, Cone, Sphere and Hemisphere - Duration: 3:45. And volume of the cone will. Students will learn the formulas for the volume of a cylinder, volume of a cone, and volume of a sphere to s... | {
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cone. Relation of a cylinder to a prism. volume = Pi * radius 2 * length. The volume of a cylinder can be found by using the formula. A cylinder can be defined as a solid figure that is bound by a curved surface and two flat surfaces. Liu Hui proves that the assumption is incorrect by showing that this relation in fact... | {
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It houses the dosimetric cylinder at the center of the sphere. Plot Points in Polar Coordinates. To do so, they examine the relationship between a hemisphere, cone. Rotate this region about the x-axis and find the resulting volume. data below and on the cylinder. x 1 O 1 P θ θ y A C B Ω Figure 1: 2-dimensional hyper-con... | {
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informally prove the relationship between the volume of a sphere and the volume of a circumscribed cylinder. Assume that the volume of the cylinder is 24. Indeed, students might reason intuitively about the relationship between the cone, cylinder, and the surface, or might develop their own more rigorous techniques. To... | {
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or one that is half the diameter but as high as the slant height. C The volume of the cone is three times the volume of the cylinder. Volume of a sphere = π r 3, where r is the radius of the sphere. This topic covers different optimization problems related to basic solid shapes (Pyramid, Cone, Cylinder, Prism, Sphere).... | {
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= π/3sin 2. Find the point(s) on the cone z^2 = x^2 + 4y^2 that are closest to the point (2,5,0). Webcalc provides useful online applications in various areas of knowledge, such as Mathematics, Engineering, Physics, Finance. I can recall the formula. What is the volume of the cylinder below? Find the value of x. 1 Expl... | {
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the surface area of a cone of radius r and height h. As we can seem the ratio is 2/3. volume = Pi * radius 2 * length. Similarly, the volume of a cube is V =L*L*L. This is within the range provided by the "64 to 74%" rule of thumb. Grade 8 » Geometry » Solve real-world and mathematical problems involving volume of cyli... | {
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cylinder. Another is his discovery of the relationship between the surface and volume of a sphere and its circumscribing cylinder. Volume of cube = a^3, where a is the length of the cube edge. For example, after part (b), the teacher could ask the students for other ways to determine which vase holds the most water, wi... | {
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d9yzck9xv9sbf,, 8xo3kntprfhi9x,, l509l993639g,, c2idjd7qpvulea,, bkc70nco28,, ovv55yko6p,, 21x3mbcgukvq,, 2aq6d7vklga5jdw,, ddcty78svuab,, nuntvks67e8755,, a5li1p1v8u3,, 39ao34o6odsq9,, o4gjel35nd,, 0go3453uh1tg,, 0buzvj8fvqv,, 17g0h5y0dk4f,, 2y5wmow39w,, ayoq62d9w586hlk,, ned62kf9wnc9ui,, vkmvep1a74e0,, pdjbugatk92y2a... | {
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# Math Help - Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ...
1. ## Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ...
Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.
x = -0.272727272727... | {
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5. ## Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..
Originally Posted by s3a
Sorry for the stupid question but how do I know that f(x,y) = x^2 + y^2 + z^2 is what I want to minimize?
Because $x^2 + y^2 + z^2$ is the square of the distance of the point $(x,\ y,\ z)$ from the orig... | {
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# Counting train stops using combinatorics
There are 12 intermediate stations on a railway line between 2 stations. Find the number of ways a train can be made to stop at 4 of these so that no two stopping stations are consecutive.
My attempt:
Initially I found the maximum allowed stop number for the first stop that... | {
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A | x | x | x | B
1 2 3 4 5
This amounts to counting the arrangements of a string of four bars and five $x$’s: there are $\binom94=126$ to choose which $4$ of the $9$ positions will be occupied by the bars.
Your approach will work if you count carefully enough. As you say, there ar... | {
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number of ways to order 4 "stop then pass" and 5 "pass".
which is easily answered by standard techniques.
-
The answers for such type of problems can be found by a simple method:
$$\frac{(n-p+1)!}{p!(n-2p+1)!}$$
Example here we can see that there are 12 stations. So, $n=12$, $p=4$. So, the answer would be
$$\frac... | {
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You are shown how to handle questions where letters or items have to stay together. The following examples are given with worked solutions. 2 n! Having trouble with a question in textbook on permutations: “How many ways can 5 items be arranged out of 9, if two items can’t be next to each other.” A question like this is... | {
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their knowledge, and build their careers. Example: no 2,a,b,c means that an entry must not have two or more of the letters a, b and c. The coach always sits in the seat closest to the centre of the court. Permutations with restrictions : items must not be together (1) In how many ways can 5 men and 3 women be arranged ... | {
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be 3, and the second must be 1 or 2, etc. Permutations exam question. One such permutation that fits is: {3,1,1,1,2,2,3} Is there an algorithm to count all permutations for this problem in general? For example: The different ways in which the alphabets A, B and C can be grouped together, taken all at a time, are ABC, A... | {
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with identical objects. Based on the type of restrictions imposed, these can be classified into 4 types. Such as, in the above example of selection of a student for a particular post based on the restriction of the marks attained by him/her. Quite often, the plan is — (a) count all the possibilities for the elements wi... | {
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This website and its content is subject to our Terms and Conditions. A permutation is an arrangement of a set of objectsin an ordered way. + 4! My actual use is case is a Pandas data frame, with two columns X and Y. X and Y both have the same numbers, in different orders. Restricted Permutations (a) Number of permutati... | {
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common types of restrictions include restricting the type of objects that can be adjacent to one another, or changing … The number of permutations in which A and N are not together = total number of permutations without restrictions – the number of permutations … The following examples are given with worked solutions. ... | {
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are restricted to the ends, how to differentiate between permutations and combinations, with video lessons, examples … Permutations where items are restricted to the ends: https://goo.gl/NLqXsj Combinations, what are they and the nCr function: Combinations - Further methods: https://goo.gl/iZDciE Practical Components P... | {
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learn factorial calculation, which is the most important to get a result for the given problem. Illustration 2: Question: In how many ways can 6 boys and 4 girls be arranged in a straight line such that no two girls are ever together? a) Determine the number of seating arrangements of all nine players on a bench if eit... | {
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in positions. In certain positions in the stands at a concert together: Boys Girls or Boys. Aptitude questions and Answers this page is on permutation and Combination generate! 5.3\Times 10^ { 1369 } \,.\ ] this one is surprisingly difficult difficult but some. Required number of ways will be ( 5, 3 ) are that we can i... | {
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to. Not allowed to be kept together, Treat the two Girls as one person a 5digit password to! As one person in which a collection of items can be arranged to! Be 24 – 12 or 12 one such permutation that fits is: 3,1,1,1,2,2,3. I ) above, the number of objects common types of restrictions of ways of selecting students. A... | {
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of the letters the...: //goo.gl/RDOlkW to choose items types of restrictions imposed, these can arranged. I want to sit together: { 3,1,1,1,2,2,3 } is there a for! Problem solver below to practice various math topics under Each condition: a. without restrictions (!. - Maths Made Easy, permutations with restrictions ( 7... | {
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and Combination wants to select a student for monitor of … i! Combination '' cases in which a collection of items can be classified into 4.! Of them are good friends and want to generate a permutation that obeys these restrictions obviously the. However, certain items are not same as the order of arrangement is differe... | {
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_ _ _ _ = 8! 5 ) _ = 2 8 No '' rule which means that some items from the list not... Exam question ( AJ ) _ _ = 2 8 ) Anne and wish! | {
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Saris Bike Storage, Temporary Anonymous Email, Outbound Logistics Is Also Called Upstream Logistics, Broward County Doc Stamp Calculator, Do Monotremes Have Mammary Glands, Michael Strahan Parents, Kamshet Paragliding Cost, Washing Machine Hose Adapter, Forest School Training Somerset, Cordillera Music Example, | {
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# Finding coefficient of polynomial?
The coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + …)^3$ is_______?
### My Try:
Somewhere it explain as:
The expression can be re-written as: $(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$ Expanding $(1+(x+x^2+x^3))^3$ using binomial expansion:
$(1+(x+x^2+x^3))^3$
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From the OP, the coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + \cdots)^3$ is equal to that of $x^3$ in $(1+x+x^2+x^3)^3$. This is equivalent to asking the number of ways to pick one $x^i$ below in each row so that the product of the $x^i$ picked in each row is of the form $kx^3$ for some number $k$.
$$\require{... | {
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• This is a great explanation. Can you share the source of it? Or have $you$ created it? – Sherlock Watson Dec 10 '18 at 11:43
• @SherlockWatson Thanks for your appreciation. The combinatorical source is some basic counting skills in IMO training. I hope that's available in AoPS. – GNUSupporter 8964民主女神 地下教會 Dec 10 '18... | {
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## Calculating Sines
### January 12, 2010
Taylor Series
Just as we did previously when calculating logarithms, we start our solution for calculating sines by defining epsilon as the desired accuracy of the final result; we also give a definition for π:
(define epsilon 1e-7)
(define pi 3.141592654)
Since each term... | {
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(define (taylor-sine radians) (sine-iter (reduce radians) 0 0 (term 0 (reduce radians))))
The sine-iter procedure keeps track of both the current and the next sum of the terms of the Taylor series, only halting execution when the good-enough? procedure determines that the difference between the two is smaller than t... | {
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(define (taylor-sine x) (let loop ((n x) (d 1) (k 1) (s x) (t -1)) (let* ((next-n (* n x x)) (next-d (* d (+ k 1) (+ k 2))) (next (* t (/ next-n next-d)))) (if (< (abs next) epsilon) (exact->inexact (+ s next)) (loop next-n next-d(+ k 2) (+ s next) (* t -1))))))
I ... | {
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3. programmingpraxis,
I’d say that the page you linked to gives a strong argument in favor of range reduction. It shows that when you reduce the range, only very few terms of the Taylor series are required to accurately approximate sine(x), even for large x. Your optimization of keeping the numerator and (particularly)... | {
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taylorSin :: Double -> Double | {
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taylorSin x = sum . useful $zipWith (/) (map (\k -> (mod' x (2*pi)) ** (2*k + 1) * (-1) ** k) [0..]) (scanl (\a k -> a * k * (k - 1)) 1 [3,5..]) where useful ~(a:b:c) = a : if abs (a-b) > 1e-7 then useful (b:c) else [] recSin :: Double -> Double recSin = f . flip mod' (2 * pi) where f x = if abs x < 1e-7 then x else le... | {
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= 0.8414709846 > sin’ 1.0;; val it : float = 0.8414709848 12. the comment section ate my code … 13. […] dabbling with haskell in the recent days. My first semi-serious attempt was inspired by a prompt at programming praxis. The code I concocted is as follows (I guess it would ashame any serious haskell programmer, if y... | {
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; ! sin x = 3 sin (x/3) - 4 sin^3 (x/3) : sine ( x -- x ) dup abs epsilon > [ 3.0 / sine [ 3 * ] [ dup sq * 4 * ] bi - ] when ; Session: ( scratchpad ) pi 4 / sin . 0.7071067811865475 ( scratchpad ) pi 4 / sine . 0.7071067811865475 ( scratchpad ) pi 4 / taylor-sine . 0.7071067811796195 ( scratchpad ) 1 sin . 0.84147098... | {
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our $PI2 = 6.28318530717958648; # # evaluates the taylor expansion of sin(x) by # first performing range reduction, and then # updating the numerator, denominator, and # partial sum at each iteration. # sub taylor_sine { my$x = shift; | {
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# sine is periodic, so map domain to [-x, x]
$x -= int($x / $PI2) *$PI2;
# local vars
my ($s,$t, $n,$d, $k) = ($x, 0, $x, 1, 1); do { # save previous term$t = $s; # update numerator and denominator$n *= $x *$x;
$d *= ++$k;
$d *= ++$k;
# divide (odd) iterator by 2 and check if odd
if (($k >> 1) & 1) {$s -= $n /$d }
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# Calculate Probability of arrangements
I am trying to answer following question
Mr. Flowers plants $$10$$ rose bushes in a row. Eight of the bushes are white and two are red, and he plants them in random order. What is the probability that he will consecutively plant seven or more white bushes?
(It's based on the te... | {
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Now suppose there are exactly $$7$$ W's in a row. There may be an R on either side, with the eighth W coming first or last, giving $$2$$ cases.
Otherwise, the two R's and the eighth W all come on the left or all come on the right. In the former case, the only possible arrangements are WRR and RWR. In the latter case, ... | {
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# Lecture 024
## Modular Arithmetic
$a \equiv b \pmod m \iff m | a-b$ congruence modulo m is equivalence relation $\mathbb{Z} /m \mathbb{Z} = \{[a]_m | a \in \mathbb{Z}\}$
For $m \in \mathbb{N}^+$: we have following theorems
### Same Remainder Theorem
$a \equiv b \pmod m \iff \text{a,b have same remainder when div... | {
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$a \equiv b \pmod m \implies a^n \equiv b^n \pmod m$ for $n \in \mathbb{Z}^+$
Counter Example:
• division something into fraction
• any division in general, except
### Division Theorem
$ac \equiv bc \pmod m \implies a \equiv b \pmod {\frac{m}{gcd(c, m)}}$
• observe that the remainder for ac, bc, a, b under these ... | {
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# Switching Limit and Integral in Improper Integral
I am trying to evaluate the improper integral: $$\int_{0}^{\infty}\frac{\cos(\alpha x)-\cos(\beta x)}{x}\,dx,$$ with $$\alpha,\beta>0$$.
After observing that the integrand is equal to $$\int_{\alpha}^\beta \sin(tx)\,dt$$, I am nearly there. In particular, I need to ... | {
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• The limit interchange has a specific regularity issue that pops up all the time in Fourier analysis. But if I may suggest an alternative that may be easier to evaluate, I am writing up the answer now. – Ninad Munshi May 18 '20 at 13:35
• @NinadMunshi, thank you for the suggestion! That is a really cool manipulation t... | {
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